Problem: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 4x + 30$, and $ m \angle BOC = 5x - 30$, find $m\angle AOB$. $O$ $A$ $C$ $B$
Solution: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {4x + 30} + {5x - 30} = {90}$ Combine like terms: $ 9x + 0 = 90$ Add $0$ to both sides: $ 9x = 90$ Divide both sides by $9$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 4({10}) + 30$ Simplify: $ {m\angle AOB = 40 + 30}$ So ${m\angle AOB = 70}$.